Explicit stability conditions for a linear trinomial delay difference equation

Abstract

We present new type of necessary and sufficient conditions for the asymptotic stability of the zero solution of the linear difference equation x ( n ) = α x ( n − m ) + β x ( n − k ) where α , β are real scalars and k > m > 0 are integers. Compared to the existing criteria, our conditions are fully explicit and simple to analyse. In particular, we show the usefulness of these conditions in the description of appropriate stability sets for the studied difference equation with fixed delays m , k as well as with fixed coefficients α , β . As a by-product, we obtain an explicit criterion guaranteeing that all the zeros of the general three-term polynomial are located inside the unit circle in the complex plane.